In addition to these we shall employ the usual symbols +, −, &c. of Algebra, and also the sign of congruence, namely ≡. This symbol has been introduced by the illustrious Gauss. BOOK I. THEORY OF ANGLES, TRIANGLES, PARALLEL LINES, AND PARALLELOGRAMS. ________________ DEFINITIONS. The Point. i. A point is that which has position but not dimensions. A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line. But a point is neither a solid, nor a surface, nor a line; hence it has no dimensions—that is, it has neither length, breadth, nor thickness. The Line. ii. A line is length without breadth. A line is space of one dimension. If it had any breadth, no matter how small, it would be space of two dimensions; and if in addition it had any thickness it would be space of three dimensions; hence a line has neither breadth nor thickness. iii. The intersections of lines and their extremities are points. iv. A line which lies evenly between its extreme points is called a straight or right line, such as AB. If a point move without changing its direction it will describe a right line. The direction in which a point moves is called its “sense.” If the moving point continually changes its direction it will describe a curve; hence it follows that only one right line can be drawn between two points. The following Illustration is due to Professor Henrici:—“If we suspend a weight by a string, the string becomes stretched, and we say it is straight, by which we mean to express that it has assumed a peculiar definite shape. If we mentally abstract from this string all thickness, we obtain the notion of the simplest of all lines, which we call a straight line.” The Plane. v. A surface is that which has length and breadth. A surface is space of two dimensions. It has no thickness, for if it had any, however small, it would be space of three dimensions. vi. When a surface is such that the right line joining any two arbitrary points in it lies wholly in the surface, it is called a plane. A plane is perfectly flat and even, like the surface of still water, or of a smooth floor.—Newcomb. Figures. vii. Any combination of points, of lines, or of points and lines in a plane, is called a plane figure. If a figure be formed of points only it is called a stigmatic figure; and if of right lines only, a rectilineal figure. viii. Points which lie on the same right line are called collinear points. A figure formed of collinear points is called a row of points. The Angle. ix. The inclination of two right lines extending out from one point in different directions is called a rectilineal angle. x. The two lines are called the legs, and the point the vertex of the angle.